Table 3.

Estimated bootstrap errors (50 and 100 calculations with random data sets) of free energy shifts calculated by the VFEP method. The system is the Na⁺:Cl⁻ system with 6 windows.

		N=50		N=100
	Window	Average	S.D.	Average	S.D.
10000 pt/w	1	0.758	0.032	0.760	0.033
	2	2.139	0.026	2.141	0.026
	3	−0.153	0.017	−0.152	0.018
	4	−0.776	0.014	−0.777	0.017
	5	−0.720	0.028	−0.722	0.026
	6	−1.249	0.033	−1.250	0.032
RMS			0.026		0.026
	Window	Average	S.D.	Average	S.D.
1000 pt/w	1	0.763	0.118	0.761	0.119
	2	2.146	0.101	2.142	0.091
	3	−0.153	0.057	−0.151	0.053
	4	−0.763	0.066	−0.771	0.065
	5	−0.722	0.093	−0.721	0.089
	6	−1.270	0.111	−1.260	0.105
RMS			0.094		0.088
	Window	Average	S.D.	Average	S.D.
100 pt/w	1	0.752	0.219	0.715	0.254
	2	2.135	0.189	2.087	0.206
	3	−0.149	0.185	−0.138	0.175
	4	−0.770	0.214	−0.761	0.181
	5	−0.685	0.236	−0.664	0.246
	6	−1.283	0.299	−1.239	0.301
RMS			0.227		0.232
	Window	Average	S.D.	Average	S.D.
20 pt/w	1	0.679	0.819	0.633	0.902
	2	2.028	0.734	1.965	0.855
	3	−0.118	0.474	−0.097	0.586
	4	−0.772	0.483	−0.728	0.54
	5	−0.654	0.652	−0.647	0.653
	6	−1.162	0.775	−1.126	0.781
RMS			0.670		0.732

The numbers here are derived from a 6-window umbrella sampling simulation on a Na⁺:Cl⁻ pair in a TIP3P water box with a biasing potential of 5 kcal/mol/Ų (see the Results section, also Figure 4). The results are estimated by performing bootstrap type error analysis on the free energy shift term, −lnZ^α, with the same calculations performed on 50 or 100 randomly chosen data sets. “S.D.” is the standard deviation while “RMS” is the root-mean-square value of the corresponding column. Results from different number of data points used in a window (10000 pt/w, 1000 pt/w, 100 pt/w, and 20 pt/w) are shown. All values are in units of k_BT.